\(\int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx\) [553]

Optimal result

Integrand size = 19, antiderivative size = 242 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac {14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac {7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac {14 \sqrt [3]{b} (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac {7 \sqrt [3]{b} (b c-a d)^2 \log \left (1-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 d^{10/3}} \] Output:

-3*(b*x+a)^(7/3)/d/(d*x+c)^(1/3)-14/3*b*(-a*d+b*c)*(b*x+a)^(1/3)*(d*x+c)^( 
2/3)/d^3+7/2*b*(b*x+a)^(4/3)*(d*x+c)^(2/3)/d^2-14/9*b^(1/3)*(-a*d+b*c)^2*a 
rctan(1/3*3^(1/2)+2/3*b^(1/3)*(d*x+c)^(1/3)*3^(1/2)/d^(1/3)/(b*x+a)^(1/3)) 
*3^(1/2)/d^(10/3)-7/9*b^(1/3)*(-a*d+b*c)^2*ln(b*x+a)/d^(10/3)-7/3*b^(1/3)* 
(-a*d+b*c)^2*ln(1-b^(1/3)*(d*x+c)^(1/3)/d^(1/3)/(b*x+a)^(1/3))/d^(10/3)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\frac {-\frac {3 \sqrt [3]{d} \sqrt [3]{a+b x} \left (18 a^2 d^2-a b d (49 c+13 d x)+b^2 \left (28 c^2+7 c d x-3 d^2 x^2\right )\right )}{\sqrt [3]{c+d x}}-28 \sqrt {3} \sqrt [3]{b} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-28 \sqrt [3]{b} (b c-a d)^2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+14 \sqrt [3]{b} (b c-a d)^2 \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{18 d^{10/3}} \] Input:

Integrate[(a + b*x)^(7/3)/(c + d*x)^(4/3),x]
 

Output:

((-3*d^(1/3)*(a + b*x)^(1/3)*(18*a^2*d^2 - a*b*d*(49*c + 13*d*x) + b^2*(28 
*c^2 + 7*c*d*x - 3*d^2*x^2)))/(c + d*x)^(1/3) - 28*Sqrt[3]*b^(1/3)*(b*c - 
a*d)^2*ArcTan[(Sqrt[3]*b^(1/3)*(c + d*x)^(1/3))/(2*d^(1/3)*(a + b*x)^(1/3) 
 + b^(1/3)*(c + d*x)^(1/3))] - 28*b^(1/3)*(b*c - a*d)^2*Log[d^(1/3)*(a + b 
*x)^(1/3) - b^(1/3)*(c + d*x)^(1/3)] + 14*b^(1/3)*(b*c - a*d)^2*Log[d^(2/3 
)*(a + b*x)^(2/3) + b^(1/3)*d^(1/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3) + b^(2 
/3)*(c + d*x)^(2/3)])/(18*d^(10/3))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {57, 60, 60, 71}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {7 b \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x}}dx}{d}-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}-\frac {2 (b c-a d) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}dx}{3 d}\right )}{d}-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}-\frac {2 (b c-a d) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{d}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}}dx}{3 d}\right )}{3 d}\right )}{d}-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 71

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}-\frac {2 (b c-a d) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{d}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}\right )}{3 d}\right )}{3 d}\right )}{d}-\frac {3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}\)

Input:

Int[(a + b*x)^(7/3)/(c + d*x)^(4/3),x]
 

Output:

(-3*(a + b*x)^(7/3))/(d*(c + d*x)^(1/3)) + (7*b*(((a + b*x)^(4/3)*(c + d*x 
)^(2/3))/(2*d) - (2*(b*c - a*d)*(((a + b*x)^(1/3)*(c + d*x)^(2/3))/d - ((b 
*c - a*d)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt 
[3]*d^(1/3)*(a + b*x)^(1/3))])/(b^(2/3)*d^(1/3))) - Log[a + b*x]/(2*b^(2/3 
)*d^(1/3)) - (3*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3 
))])/(2*b^(2/3)*d^(1/3))))/(3*d)))/(3*d)))/d
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 71

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( 
Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + 
 b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {7}{3}}}{\left (x d +c \right )^{\frac {4}{3}}}d x\]

Input:

int((b*x+a)^(7/3)/(d*x+c)^(4/3),x)
 

Output:

int((b*x+a)^(7/3)/(d*x+c)^(4/3),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (188) = 376\).

Time = 0.11 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=-\frac {28 \, \sqrt {3} {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} d \left (-\frac {b}{d}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + 14 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 28 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 3 \, {\left (3 \, b^{2} d^{2} x^{2} - 28 \, b^{2} c^{2} + 49 \, a b c d - 18 \, a^{2} d^{2} - {\left (7 \, b^{2} c d - 13 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (d^{4} x + c d^{3}\right )}} \] Input:

integrate((b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="fricas")
 

Output:

-1/18*(28*sqrt(3)*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b* 
c*d^2 + a^2*d^3)*x)*(-b/d)^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x + a)^(1/3)*(d* 
x + c)^(2/3)*d*(-b/d)^(2/3) + sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)) + 14*( 
b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x) 
*(-b/d)^(1/3)*log(((d*x + c)*(-b/d)^(2/3) - (b*x + a)^(1/3)*(d*x + c)^(2/3 
)*(-b/d)^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) - 28*(b^2*c^3 
 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x)*(-b/d) 
^(1/3)*log(((d*x + c)*(-b/d)^(1/3) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x 
 + c)) - 3*(3*b^2*d^2*x^2 - 28*b^2*c^2 + 49*a*b*c*d - 18*a^2*d^2 - (7*b^2* 
c*d - 13*a*b*d^2)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d^4*x + c*d^3)
 

Sympy [F]

\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{3}}}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \] Input:

integrate((b*x+a)**(7/3)/(d*x+c)**(4/3),x)
 

Output:

Integral((a + b*x)**(7/3)/(c + d*x)**(4/3), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(7/3)/(d*x + c)^(4/3), x)
 

Giac [F]

\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(7/3)/(d*x + c)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \] Input:

int((a + b*x)^(7/3)/(c + d*x)^(4/3),x)
 

Output:

int((a + b*x)^(7/3)/(c + d*x)^(4/3), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx=\left (\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} c +\left (d x +c \right )^{\frac {1}{3}} d x}d x \right ) a^{2}+\left (\int \frac {\left (b x +a \right )^{\frac {1}{3}} x^{2}}{\left (d x +c \right )^{\frac {1}{3}} c +\left (d x +c \right )^{\frac {1}{3}} d x}d x \right ) b^{2}+2 \left (\int \frac {\left (b x +a \right )^{\frac {1}{3}} x}{\left (d x +c \right )^{\frac {1}{3}} c +\left (d x +c \right )^{\frac {1}{3}} d x}d x \right ) a b \] Input:

int((b*x+a)^(7/3)/(d*x+c)^(4/3),x)
 

Output:

int((a + b*x)**(1/3)/((c + d*x)**(1/3)*c + (c + d*x)**(1/3)*d*x),x)*a**2 + 
 int(((a + b*x)**(1/3)*x**2)/((c + d*x)**(1/3)*c + (c + d*x)**(1/3)*d*x),x 
)*b**2 + 2*int(((a + b*x)**(1/3)*x)/((c + d*x)**(1/3)*c + (c + d*x)**(1/3) 
*d*x),x)*a*b